Embedded newform 245.2.x.a.103.5

• Label: 245.2.x.a.103.5
• Level: $245$
• Weight: $2$
• Character: 245.103
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.5
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17937 + 1.59799i) q^{2} +(0.467157 - 0.0883908i) q^{3} +(-0.573145 - 1.85809i) q^{4} +(-1.51428 - 1.64528i) q^{5} +(-0.409703 + 0.850757i) q^{6} +(1.41438 - 2.23596i) q^{7} +(-0.104082 - 0.0364200i) q^{8} +(-2.58220 + 1.01344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17937	+	1.59799i	−0.833940	+	1.12995i	0.155902	+	0.987773i	\(0.450172\pi\)
							−0.989841	+	0.142176i	\(0.954590\pi\)
\(3\)	0.467157	−	0.0883908i	0.269713	−	0.0510325i	−0.0492957	−	0.998784i	\(-0.515698\pi\)
							0.319009	+	0.947752i	\(0.396650\pi\)
\(5\)	−1.51428	−	1.64528i	−0.677206	−	0.735793i
							\(7\)	1.41438	−	2.23596i	0.534586	−	0.845114i
\(11\)	2.24757	−	5.72672i	0.677669	−	1.72667i								−0.00927030	−	0.999957i	\(-0.502951\pi\)
							0.686939	−	0.726715i	\(-0.258954\pi\)
\(13\)	−2.37121	−	0.267172i	−0.657656	−	0.0741001i	−0.223172	−	0.974779i	\(-0.571641\pi\)
							−0.434485	+	0.900679i	\(0.643070\pi\)
\(17\)	−2.24009	−	0.0838184i	−0.543302	−	0.0203289i	−0.235668	−	0.971834i	\(-0.575728\pi\)
							−0.307634	+	0.951505i	\(0.599537\pi\)
\(19\)	1.77835	−	3.08020i	0.407982	−	0.706646i	−0.586681	−	0.809818i	\(-0.699566\pi\)
							0.994664	+	0.103172i	\(0.0328992\pi\)
\(23\)	0.116294	+	3.10801i	0.0242489	+	0.648066i	0.957431	+	0.288662i	\(0.0932104\pi\)
							−0.933182	+	0.359404i	\(0.882980\pi\)
\(29\)	−1.81586	+	0.414458i	−0.337196	+	0.0769629i	−0.387766	−	0.921758i	\(-0.626753\pi\)
							0.0505701	+	0.998721i	\(0.483896\pi\)
\(31\)	6.67926	−	3.85627i	1.19963	−	0.692607i	0.239157	−	0.970981i	\(-0.423129\pi\)
							0.960473	+	0.278374i	\(0.0897955\pi\)
\(37\)	2.98158	−	1.57581i	0.490169	−	0.259062i	−0.203872	−	0.978998i	\(-0.565352\pi\)
							0.694041	+	0.719936i	\(0.255829\pi\)
\(41\)	−0.626876	−	1.30172i	−0.0979016	−	0.203295i	0.846266	−	0.532761i	\(-0.178846\pi\)
							−0.944167	+	0.329466i	\(0.893131\pi\)
\(43\)	1.96278	+	5.60929i	0.299320	+	0.855409i	0.991079	+	0.133279i	\(0.0425507\pi\)
							−0.691758	+	0.722129i	\(0.743164\pi\)
\(47\)	−5.99749	−	4.42635i	−0.874824	−	0.645650i	0.0609920	−	0.998138i	\(-0.480574\pi\)
							−0.935816	+	0.352489i	\(0.885336\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.5	✓	624
5.2	odd	4	inner	245.2.x.a.152.5	yes	624
49.10	odd	42	inner	245.2.x.a.108.5	yes	624
245.157	even	84	inner	245.2.x.a.157.5	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.5	✓	624	1.1	even	1	trivial
245.2.x.a.108.5	yes	624	49.10	odd	42	inner
245.2.x.a.152.5	yes	624	5.2	odd	4	inner
245.2.x.a.157.5	yes	624	245.157	even	84	inner